In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),^[1][2] is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them.

Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.^[3][4][5]

If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.^[3][4][5]

The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.^[6][7][8]

The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.^[9]

Overview

Notation

Some textbooks have the ones on the subdiagonal; that is, immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal.^[10][11]

Motivation

An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following matrix:

${\displaystyle A=\left{\begin{array}{*{20}{r}}5&4&2&1\\2pt0&1&-1&-1\\2pt-1&-1&3&0\\2pt1&1&-1&2\end{array}}\right.}$

Including multiplicity, the eigenvalues of A are λ = 1, 2, 4, 4. The dimension of the eigenspace corresponding to the eigenvalue 4 is 1 (and not 2), so A is not diagonalizable. However, there is an invertible matrix P such that J = P⁻¹AP, where

${\displaystyle J={\begin{bmatrix}1&0&0&0\\2pt0&2&0&0\\2pt0&0&4&1\\2pt0&0&0&4\end{bmatrix}}.}$

The matrix ${\displaystyle J}$ is almost diagonal. This is the Jordan normal form of A. The section Example below fills in the details of the computation.

Complex matricesedit

In general, a square complex matrix A is similar to a block diagonal matrix

${\displaystyle J={\begin{bmatrix}J_{1}&\;&\;\\\;&\ddots &\;\\\;&\;&J_{p}\end{bmatrix}}}$

where each block J_i is a square matrix of the form

${\displaystyle J_{i}={\begin{bmatrix}\lambda _{i}&1&\;&\;\\\;&\lambda _{i}&\ddots &\;\\\;&\;&\ddots &1\\\;&\;&\;&\lambda _{i}\end{bmatrix}}.}$

So there exists an invertible matrix P such that P⁻¹AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each J_i is called a Jordan block of A. In a given Jordan block, every entry on the superdiagonal is 1.

Assuming this result, we can deduce the following properties:

• Counting multiplicities, the eigenvalues of J, and therefore of A, are the diagonal entries.
• Given an eigenvalue λ_i, its geometric multiplicity is the dimension of ker(A − λ_i I), where I is the identity matrix, and it is the number of Jordan blocks corresponding to λ_i.^[12]
• The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λ_i is its algebraic multiplicity.^[12]
• A is diagonalizable if and only if, for every eigenvalue λ of A, its geometric and algebraic multiplicities coincide. In particular, the Jordan blocks in this case are 1 × 1 matrices; that is, scalars.
• The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as N_ij = δ_i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the exponential exp(A).
• The number of Jordan blocks corresponding to λ_i of size at least j is dim ker(A − λ_iI)^j − dim ker(A − λ_iI)^j−1. Thus, the number of Jordan blocks of size j is

  ${\displaystyle 2\dim \ker(A-\lambda _{i}I)^{j}-\dim \ker(A-\lambda _{i}I)^{j+1}-\dim \ker(A-\lambda _{i}I)^{j-1}}$

• Given an eigenvalue λ_i, its multiplicity in the minimal polynomial is the size of its largest Jordan block.

Exampleedit

Consider the matrix ${\displaystyle A}$ from the example in the previous section. The Jordan normal form is obtained by some similarity transformation:

${\displaystyle P^{-1}AP=J;}$ that is, ${\displaystyle AP=PJ.}$

Let ${\displaystyle P}$ have column vectors ${\displaystyle p_{i}}$, ${\displaystyle i=1,\ldots ,4}$, then

Zdroj:https://en.wikipedia.org?pojem=Jordan_normal_form
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